Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands

Moiré superlattices engineer band properties and enable observation of fractal energy spectra of Hofstadter butterfly. Recently, correlated-electron physics hosted by flat bands in small-angle moiré systems has been at the foreground. However, the implications of moiré band topology within the single-particle framework are little explored experimentally. An outstanding problem is understanding the effect of band topology on Hofstadter physics, which does not require electron correlations. Our work experimentally studies Chern state switching in the Hofstadter regime using twisted double bilayer graphene (TDBG), which offers electric field tunable topological bands, unlike twisted bilayer graphene. Here we show that the nontrivial topology reflects in the Hofstadter spectra, in particular, by displaying a cascade of Hofstadter gaps that switch their Chern numbers sequentially while varying the perpendicular electric field. Our experiments together with theoretical calculations suggest a crucial role of charge polarization changing concomitantly with topological transitions in this system. Layer polarization is likely to play an important role in the topological states in few-layer twisted systems. Moreover, our work establishes TDBG as a novel Hofstadter platform with nontrivial magnetoelectric coupling.

1) The authors report magneto-transport studies of twisted double bilayer graphene. Similar works have been reported (ref.11-15 in the manuscript). Since the manuscript does not report original experiments, the authors should clarify the progress made here.
2) In the opening bold paragraph, the authors claim that "However, the topological properties of these moire bands such as Chern numbers are little explored". This is not true (see refs.18-26 and many more). In fact, it well known that topology of bands is very import for understanding moire systems. The following recent paper is directly relevant to the specific system studied here, 3) For twisted double bilayer graphene, there is strong evidence for spin-polarized state. The authors should address how these new states modify Hofstadter bands, if they are still relevant. 4) As for the "Electric field drives Chern transition in Hoftstader bands" and "...switch their Chern number on at a time as we vary the electric field". Perhaps the author should mark the Chern number in Fig.4 as function of D. 5) All the experimental data supporting Chern gap is a dip in magnetoresistance. It is common practice in the field to have additional confirmation, such as activation measurement that gives estimate of gap size, especially considering this is a follow-up study.
In conclusion, the experiments appear not original, and I do not find the "new" interpretation of the data convincing or sound. I cannot recommend its publication.
Reviewer #2 (Remarks to the Author): The manuscript of Adak et al. describes a magnetotransport study in twisted double bilayer graphene with twist angles of ~1deg. The study is focused on the evolution of quantum Hall gaps with different Chern numbers as a function of transverse electric field. The study combines theory and experiment and the theoretical and experimental results are in agreement. The main result is that at a fixed electric field a subset of gaps with certain Chern numbers are more prominent. The study is carefully carried out, and the manuscript accessible to a broad audience.
There are aspects of the manuscript that could desirably be revised or augmented for publication. The narrative invokes the non-trivial band topology rather often, but the result do not clearly highlight it. How would the experimental observations change if e.g. the bands had the same dispersion, but different topology? A good comparison are twisted double bilayer graphene with twist angles close to 180deg. Figure 2a and 2c data are plotted in a way that makes it difficult for the reader to get much out of it, except for the gaps already marked.
Reviewer #3 (Remarks to the Author): The authors report their work on 'Electric field drives Chern transition in Hofstadter bands of twisted double bilayer graphene', and try to understand the effect of band topology on Hofstadter physics. Revealing the influence of band topology on the correlation effects and Hofstadter Physics is interesting and important in current twisted moire system studies. However, after go over the whole manuscript, I worry that the conclusion made by the authors could not be well supported by the present data. So, I would not recommend it to be published in Nature Communications unless the authors could provide further critical experimental supports.
Here are several comments： 1, According to previous transport measurements on either twisted bilayer graphene or monobilayer graphene, the nontrivial topological Chern bands have been unveiled near the integer fillings. The spontaneous symmetry broken plays an important role there. The authors studied twisted double bilayer graphene, and also finds the Chern bands emanating from the integer fillings, personally, I do not see any progress or advance from the current work as comparing to those work. The authors must address the difference.
2, In Fig.2a and Fig.2b, the authors try to show the displacement field driven the Chern number variation. In order to confirm the readers, the authors use some lines to guide the eyes. However, from the present data, there are several choices on draw the line. For example, for D~-0.02, near the (0,0), there are extra fan structure flank the Dirac point with positive and negative slope, maybe give (2,0) or (-2,0). Similar operation could be done on larger D. So I do not confirmed that there is an electric field driven Chern number transition, at least by the current presented data.
3, based on question 2, even in other twisted graphene system with magnetic field and strong correlations, similar behavior could be found. I am not confirmed it its unique to twisted double bilayer graphene, and limited to band topology, unless the band topology here means the strong electron interaction driven nontrivial band topology, then it has been well studied in other systems. 4, In Fig.2, the authors try to use the Fan expanding to other bands to claim the band topology contribution, but the signal or evidence is not that solid, especially under the current data quality. In order to claim this, the authors must improve the data or sample quality. 5, I am not quite sure how well of the current data could be well repeated in other twisted angles. How about the twisted double bilayer near the right angle, like 1.2-1.4degreee? There the physics should be clearer especially the presented understanding would base on an insulating state on the integer fillings. Currently, the authors need a magnetic field to trigger such an insulating gap.

Work
Focus of the work G W Burg et al, Correlated Insulating States in Twisted Double Bilayer Graphene. Phys. Rev. L 123, 197702 (2019).
• Reports correlated insulators states at ½ filling at zero magnetic field, at ⅓ and ¾ at finite parallel magnetic fields. • Suggests the role of band separation along with band flatness to observe correlated gaps.
• Reports control of electric field over the correlated gaps.
• The measurement under a parallel magnetic field suggests spin-polarized ordering.
P C Adak et al, Tunable bandwidths and gaps in twisted double bilayer graphene on the verge of correlations. Phys. Rev. B 101, 125428 (2020).
• Explores band properties from temperature-dependent resistivity. • Quantifies electric field tunable bandgap magnitudes and bandwidths.
• Reports correlated insulator states that are highly sensitive to the electric field as well as the twist angle. • Provides evidence of spin-polarized ground states.
• Reports tunable correlated gaps in higher moiré bands.
• Demonstrates a flat band with supporting correlated insulating states that are tunable by the perpendicular electric field. • Reports a phase transition from a normal metal to a spinpolarized correlated state at half-filling.

This work
• Reports electric field tunable Chern gaps in the Hofstadter bands. • Study role of the electric field tunable layer polarization and the topology on Hofstadter spectra.  Action: To clarify the progress made in our work further, we have modified the introductory paragraphs in the revised manuscript. In particular, we noted, "While the earlier experiments in 3 of 16 TDBG have a major focus on electron correlation physics, the tunability of the topological flat bands in the Hofstadter regime is little explored." To reflect that we have further understood the mechanism of electric field tunability in terms of layer polarization changes, we have modified the title to be "Vertical electric field drives Chern transitions and layer polarization changes in Hofstadter bands".
2) In the opening bold paragraph, the authors claim that "However, the topological properties of these moire bands such as Chern numbers are little explored". This is not true (see refs.18-26 and many more). In fact, it well known that topology of bands is very import for understanding moire systems. The following recent paper is directly relevant to the specific system studied here, Phase diagram and orbital Chern insulator in twisted double bilayer graphene Yi-Xiang Wang, Fuxiang Li, and Zi-Yue Zhang Phys. Rev. B 103, 115201 -Published 1 March 2021 Reply: We thank the reviewer for the question and for pointing out the reference, which we have now cited in the revised version. At the outset, we would like to distinguish our work from the past that we combine detailed experiments with our theoretical calculations to confirm our observations. Furthermore, as Reviewer 2 assesses, "the theoretical and experimental results are in agreement." While there are some important theoretical works that explore the Chern physics in TDBG, no experimental study of TDBG has focused on this physics so far. Indeed, the rich physics of tunable topology explored in these theoretical works motivate us to experimentally demonstrate the capability of tuning Chern gaps by the electric field. Our work is also timely, as there has been a rapidly growing interest in exploring correlated Chern insulator states in other twisted graphene systems such as twisted bilayer graphene. Since the Chern insulator states explored in TBG are not tunable by the perpendicular electric field, TDBG provides a platform for engineering such topological states.

Action:
To remove any potential confusion we have modified the statement in the opening bold paragraph to highlight that our work explores Chern transition physics experimentally, not only just theoretically. To contrast our work from the experiments exploring correlated Chern insulators in TBG, we have included these sentences in the second paragraph of the revised manuscript: "In the physics of Chern insulator states, as also in the quantum Hall physics due to the formation of Landau levels, gaps with different Chern numbers can be accessed by changing the Fermi energy by varying the charge density. In contrast, a pure electrical control, such as the perpendicular electric field, to manipulate the Chern states has not been demonstrated yet." We have also included the reference suggested by the reviewer in the revised manuscript.
3) For twisted double bilayer graphene, there is strong evidence for spin-polarized state. The authors should address how these new states modify Hofstadter bands, if they are still relevant.

Reply:
We thank the reviewer for the thoughtful question. As we discussed in the main text, we find that Chern gaps change sequentially by 1. To reproduce the experimental results we need to 4 of 16 consider a large spin-splitting in our theoretical calculation ( = 2.5 meV). The spin-spitting is exchange-enhanced, most likely due to the underlying ferromagnetic order of the spin-polarized states. This supports the statement of the reviewer that there is a spin-exchange interaction in TDBG even though there is no fully formed = 2 gap in our device with a twist angle of 1.10 degrees.
To address the query of the reviewer, we have now measured a new device with a twist angle of 1.46 degrees where we observe a fully formed = 2 correlated gap that separates the spinpolarized states around | | ∼ 0.3 − 0.4 V/nm even at a zero magnetic field (see Fig. R1a). In this device also, we see vs. D plot with Chern number transition in a sequence of 1 -consistent with our earlier device; this highlights the relevance of underlying spin-polarization in enhancing spinsplitting in the Hofstadter subbands. The value of indicates that there is spin-exchange interaction even before the fully formed = 2 gap is manifested. Fig. R1: a, Conductance as a function of moiré filling factor ( ) and displacement field ( ) for a device with twist angle of 1.46 degrees. Correlated insulator gaps are observed at = 2 around | | ∼ 0.3 − 0.4a. b, A fan diagram at an electric field where the correlated gap is observed. 4) As for the "Electric field drives Chern transition in Hoftstader bands" and "...switch their Chern number on at a time as we vary the electric field". Perhaps the author should mark the Chern number in Fig.4

as function of D.
Reply: We thank the reviewer for the comment. The physics of switching the Chern number one at a time as we vary the electric field is explored in detail in Fig. 2a. We have also marked the corresponding Chern numbers by different colored lines as indicated in the legend. We reproduce this in Fig. R2a.
In Fig. 4 of the previous manuscript (reproduced in Fig. R2b), we focused on the variation of conductance at a fixed value of = 0. While this plot is not useful to extract the Chern number of the underlying zero energy bands, this allows us to explore the physics of multiple closings and reopenings of gaps governed by tunable layer polarization. We have verified the physics in a new device (see Fig. R2c) as well. However, we realized that this part could potentially distract the reader from the main physics we explore and hence moved it to the Supplementary Information in the revised version.

5) All the experimental data supporting Chern gap is a dip in magnetoresistance.
It is common practice in the field to have additional confirmation, such as activation measurement that gives estimate of gap size, especially considering this is a follow-up study.

Reply:
We thank the reviewer for this important question. However, we would like to point out that we had already confirmed the existence of the Chern gap by activation measurement and provided the values of the most prominent gaps in Fig. 2d of the last submitted version of the Supplementary Information (also reproduced in Fig. R3). We find the most prominent gaps to have a value of 0.12-0.42 meV. In conclusion, the experiments appear not original, and I do not find the "new" interpretation of the data convincing or sound. I cannot recommend its publication.

Reply:
We thank the reviewer for the comment. However, we believe that our response to the reviewer's questions will convince the new physics found in our work, namely, the experimental observation of electric field tunable Chern gaps in TDBG; this has not been reported in any other 2D systems. In addition, our theoretical calculations explain our experimental work and provide insights into the role of layer polarization. We have provided additional data from new measurements in the revised Supplementary Information to further support our claims.

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Reviewer #2: Reply: We thank the reviewer for nicely summarizing our work. We appreciate the reviewer's assessment that "the study is carefully carried out", and highlighting that our "theoretical and experimental results are in agreement". We also thank the reviewer for noting that "the manuscript is accessible to a broad audience".
There are aspects of the manuscript that could desirably be revised or augmented for publication.

Reply:
We thank the reviewer for the suggestion to compare two types of twisted double bilayer graphene systems: AB-AB and AB-BA (twisted with an angle close to 180 degrees). We have measured new devices with the AB-BA configuration. In Fig. R4, we show fan diagrams for two devices with the two different configurations at finite electric fields. At a finite electric field, the CNP gap has a nonzero valley Chern number of 2 in the AB-AB case, whereas for AB-BA the CNP gap has a zero Chern number [Koshino, Phys. Rev. B 99, 235406 (2019)]. Interestingly, we find from Fig. R4 that the CNP gap closes at / 0 ∼ 1/5 in the AB-AB system, in contrast to the CNP gap in the AB-BA device, which remains open throughout the range of magnetic fields we measured. This is consistent with the fact that two bands separated by a nontrivial gap of Chern number have the Hofstadter spectra connected by a gap closing at / 0 < 1/| |, whereas the Hofstadter spectra of two bands separated by a trivial gap can remain separated for any flux. The difference in the evolution of the CNP gap with the magnetic field for two different configurations of AB-AB and AB-BA is also reflected in Fig. R5, where we plot the conductance at = 0 as a function of and . For AB-AB, the conductance at = 0 remains low at high magnetic fields indicating a gap. This is consistent with zero valley Chern number at the CNP gap for AB-AB TDBG at zero electric field. In contrast, for AB-BA, the trend is the opposite, i.e., the conductance is higher around = 0 throughout the range of the magnetic field we measure, indicating gap closing. This is consistent with the nonzero valley Chern number at the CNP gap for AB-BA configuration at zero electric field.  We also performed detailed calculations to compare the AB-AB and the AB-BA TDBG with the same twist angle of 1.10 degrees at the interlayer potential of = 20 meV (see Fig. R6). While the band structures are quite similar (see Fig. R6a,b), the Chern numbers are different, as correctly pointed out by the reviewer. In Figs. R6c and R6d, we have plotted the corresponding evolution of Hofstadter gaps as a function of and , and we see a prominent difference between the two cases. To point out the role of topology clearly, we have plotted the evolution of Hofstadter gaps as a function of and the magnetic flux / 0 at = 20 meV in Figs. R6e and R6f for AB-AB and AB-BA, respectively. Again, we find that the CNP gap in the Hofstadter spectra is open throughout the range 0 < / 0 < 1 for AB-BA, whereas the gap closes at / 0 < 0.5 for AB-AB. This is consistent with the valley Chern numbers = 2 for AB-AB and = 0 for AB-BA at the CNP gap at a finite electric field. Action: In Fig. 4 of the revised manuscript, we have a more focused discussion on the role of topology in the Hofstadter spectra of TDBG. We have further added the calculation of Hofstadter spectra for AB-BA configuration in the revised Fig. 4.

Figure 2a and 2c data are plotted in a way that makes it difficult for the reader to get much out of it, except for the gaps already marked.
Reply: We thank the reviewer for the comment. Our central result that the Chern gaps in the Hofstadter bands of TDBG are tunable with the electric field is true for all the Chern gaps we observe; we want to emphasize that this is not an effect that can be realized by tuning density or doping. However, to avoid distraction we have focused our analysis on the pair of larger gaps in Figs

Action:
We have added the plot with more Chern numbers marked in the revised Supplementary Information. We have provided similar analysis for other twist angles as well from new measurements in the revised Supplementary Information.

The authors report their work on 'Electric field drives Chern transition in Hofstadter bands of twisted double bilayer graphene', and try to understand the effect of band topology on Hofstadter physics. Revealing the influence of band topology on the correlation effects and Hofstadter
Physics is interesting and important in current twisted moire system studies. However, after go over the whole manuscript, I worry that the conclusion made by the authors could not be well supported by the present data. So, I would not recommend it to be published in Nature Communications unless the authors could provide further critical experimental supports.

Reply:
We thank the reviewer for valuable comments and for pointing out the importance of understanding the influence of band topology in twisted moiré systems. We have now performed more experimental measurements on new devices as well as obtained further numerical data to support our claim. The central result that in TDBG the perpendicular electric field changes the Chern gaps is verified for different twist angles across multiple devices.
Here are several comments： 1, According to previous transport measurements on either twisted bilayer graphene or monobilayer graphene, the nontrivial topological Chern bands have been unveiled near the integer fillings. The spontaneous symmetry broken plays an important role there. The authors studied twisted double bilayer graphene, and also finds the Chern bands emanating from the integer fillings, personally, I do not see any progress or advance from the current work as comparing to those work. The authors must address the difference.

Reply:
We thank the reviewer for the important question that helps us to clearly differentiate our work from existing literature. As the reviewer has correctly pointed out, some recent works (e.g., Refs 13-18 of the revised manuscript) on twisted bilayer graphene (TBG) or mono-bilayer graphene have reported correlated Chern insulators emanating from integer fillings. These gaps are mostly understood as originating from the two sets of four-fold degenerate (i.e., total eight) low-energy flat bands with non-zero Chern numbers (see Fig. R8a). As the reviewer has correctly pointed out, the main role to separate the eight Chern bands is played by spontaneous symmetry breaking due to electron correlations -the flat band is important for realizing the physics. While the magnetic field helps to unveil the physics, the Hofstadter physics is not essential in understanding the origin of those Chern insulator states.
However, in our work we explore Chern gaps owing to Hofstadter physics at high enough magnetic fields such that / 0 ∼ 1 (see Fig. R8b). While correlations in flat bands can play a secondary role in enhancing the gaps, the main role is played by the interplay of the magnetic length scale and the moiré periodicity to induce fractal-like energy spectra. The signature of the Hofstadter physics is prominent in our data as we also observe strong Brown-Zak oscillations. The Hofstadter physics itself does not require the underlying band to be flat. Indeed, our results of electric field tunable Chern gaps in the Hofstadter band are observed for a wide range of twist While in both cases Chern insulator gaps are observed, the main novelty of our work is that we can use the perpendicular electric field to change Chern numbers in TDBG. We emphasize that this is different from accessing different Landau levels by changing doping in graphene-based systems. Indeed, the pure electric field control over Chern gaps in the Hofstadter bands has not been demonstrated in any other system in the existing literature. Unlike in TDBG, this is not possible in TBG since the band structure in TBG cannot be tuned by a perpendicular electric field and this is why experimental works in TBG thus far employ single gates, while in TDBG devices one uses dual gates to independently control the charge density and the electric field. Action: To further clarify the progress made in our work we have added a few sentences in the introductory paragraph of the revised manuscript. In particular, we have noted: "In the physics of Chern insulator states, as also in the quantum Hall physics due to the formation of Landau levels, gaps with different Chern numbers can be accessed by changing the Fermi energy by varying the charge density. In contrast, a pure electrical control, such as the perpendicular electric field, to manipulate the Chern states has not been demonstrated yet." 2, In Fig.2a  Reply: We thank the reviewer for the important question. We agree that the data indeed shows many gaps which naturally occur in the fractal structure of the Hofstadter spectra. In Fig. R9, we have identified different gaps with different colors. Our most interesting observation is that the position of gaps with similar magnitude changes on the -axis as is changed: see the set of gaps marked by the same color. This means that the Chern numbers of the gaps change with the electric field. We then focus on two sets of larger gaps, for sake of brevity, extract their Chern numbers, and see how those numbers evolve with the magnetic field. For reference, in Fig. R9, we have marked the other Chern gaps as well.
Similar observations have been made from the data for different twist angles also, as we explained later (in the reply to comment 5 on page 16). 3, based on question 2, even in other twisted graphene system with magnetic field and strong correlations, similar behavior could be found. I am not confirmed it its unique to twisted double bilayer graphene, and limited to band topology, unless the band topology here means the strong electron interaction driven nontrivial band topology, then it has been well studied in other systems.

Reply:
We thank the reviewer for the question. While we have pointed out the difference between the physics we study in our work and those in other twisted graphene systems in the reply to question 1 on page 11, here we would also like to reiterate the uniqueness of twisted double bilayer graphene. The band structure and the topology in TDBG are tunable by the electric field, whereas the electric field does not play any significant role in twisted bilayer graphene. Fig.2, the authors try to use the Fan expanding to other bands to claim the band topology contribution, but the signal or evidence is not that solid, especially under the current data quality.

4, In
In order to claim this, the authors must improve the data or sample quality.
Reply: To address the reviewer's comment we have made new devices using the state-of-the-art fabrication method (e.g., using graphite local gate) and reproduced our central result of changing Chern numbers of the Hofstadter gaps by the electric field across multiple angles. However, for the higher angles, the crossing point as evidence of the fan expanding to other bands is not accessible with the magnetic field available to us. To provide further support to our claim, we have done more theoretical calculations.
In Fig. R10a, we show the Hofstadter energy spectrum at the interlayer potential of = 20 meV calculated for 1.1 degrees twisted AB-AB TDBG. The corresponding evolution of the gaps is shown in Fig. R10b. For a finite electric field, the CNP gap ( = 0) of the zero magnetic field band structure of AB-AB TDBG has a valley Chern number of 2. From Fig. R10a-b, we find that the CNP gap under a perpendicular magnetic field closes at / 0 < 0.5. This is consistent with the fact that two bands separated by a nontrivial gap of Chern number have the Hofstadter spectra connected by a gap closing at / 0 < 1/| |. In contrast, the Hofstadter spectra of two bands separated by a trivial gap can remain separated for any flux. In Fig. R10c-d, we show the corresponding result for AB-BA configuration with the same twist angle and the interlayer potential from our new calculation. We find that the CNP gap remains open throughout the whole range of 0 < / 0 < 1, consistent with the zero valley Chern number of AB-BA TDBG under a finite electric field.
To highlight the signature of topology in our experimental data, we plot a fan diagram showing as a function of and in Fig. R10e for a TDBG device with a twist angle of 1.1 degrees. We see that the moiré gap (0,4) is dominated by the Hofstadter gap (12,0) at / 0 = 1/3. To clearly visualize this effect, we have plotted a line slice of at the moiré gap as a function of and see a dip at the crossing point between two gaps. The dominance of the Hofstadter gap over the moiré gap is also visualized from the color-scale plot of , where we see a finite at the crossing point, whereas is otherwise zero at the moiré gap. These results establish that the Hofstadter spectra of the two bands separated by the moiré gap are connected -a manifestation of the nontrivial topology. Action: To further corroborate our experimental observation of the signature of nontrivial band topology on the Hofstadter spectra, we have provided new calculations in Fig. 4 of the revised manuscript. Also, to visualize the signature of topology in our experimental data more clearly, we have provided the color-scale plot of along with the line slice in the revised Fig. 4.

5, I am not quite sure how well of the current data could be well repeated in other twisted angles.
How about the twisted double bilayer near the right angle, like 1.2-1.4degreee? There the physics should be clearer especially the presented understanding would base on an insulating state on the integer fillings. Currently, the authors need a magnetic field to trigger such an insulating gap.

Reply:
We thank the reviewer for asking this question. As per the suggestion of the reviewer, we verified the repeatability of our results for different twist angles by making new devices. In Fig.  R11a, we show the evolution of conductance as a function of the filling factor and the electric field for a twist angle of 1.46 degrees. Indeed, for this twist angle, we see a clear correlated gap at = 2 at the zero magnetic field (see Fig. R11b for a line slice). Fig. R11c shows the evolution of conductance at 12.5 T. Again, we observe multiple Hofstadter gaps that change their Chern numbers with the perpendicular electric field. Similar observations have been made for other twist angles as well as included in the revised Supplementary Information.

Action:
We have added the results from new measurements for different twist angles in the revised Supplementary Information. This establishes the repeatability of our observation even for the angles that the reviewer points out where an explicit = 2 gap is seen at the zero magnetic field.

Reviewers' comments:
Reviewer #1 (Remarks to the Author): The authors have made significant revisions to the manuscript and added new experimental data. Their responses to reviewers' comments are reasonable. Now I recommend its publication in Nature Communications.
Reviewer #2 (Remarks to the Author): The revised manuscript includes some improvements from the previous version. However, the experimental data remains perhaps not entirely compelling. The main finding is the observation of a set of quantum Hall states (QHSs) which originate at half moire Brillouin zone (BZ) filling (s=-2), which evolve with the transverse electric field. The authors explain theoretically the findings using a model where the both the valley and spin degrees or freedom are no longer degenerate. The valleys are split by the electric field, and the spin through the use of a Zeeman splitting in the model. The calculations identify gaps at fixed energy that change Chern number as the electric field is varied.
However the identification of the s=-2 QHS is not too convincing. The lines marking the various s=-2 QHSs in Fig. 2b are fitted over a narrow range of magnetic field, and the slope may have uncertainty. In some cases, it's not clear that a line can be drawn with any accuracy through the data, as it's the case for (1,-2) state in Fig. 2b. Moreover, a distinctive element in the data is that most s=-2 QHS have positive Chern numbers, while previous studies in TDBG (arXiv:2109.08255) or TBG observe QHSs with primarily, though not exclusively, negative Chern numbers for s<0, and positive Chern numbers for s>0. Lastly, based on what is known on the electrostatics of TDBG (DOI:10.1126/science.abc3534) the range of on-site layer energies used in the simulations appears high compared to the electric field used experimentally.
Some of the concluding statements are not entirely consistent with the findings. For examples the authors state "In the physics of Chern insulator states, as also in the quantum Hall physics due to the formation of Landau levels, gaps with different Chern numbers can be accessed by changing the Fermi energy by varying the charge density. In contrast, a pure electrical control, such as the perpendicular electric field, to manipulate the Chern states has not been demonstrated yet." The data shows that one have to change both the electric field and the density in order to change the Chern number. In the calculations the Chern gaps evolve with electric field at a fixed energy. In the experimental data at a fixed magnetic field (flux/moire unit cell) QHS with different Chern numbers are observed at different electric field AND different moire Brillouin zone filling (\nu).
Overall the findings are interesting and probably correct, although the experimental data is not entirely convincing.
Reviewer #3 (Remarks to the Author): I would appreciate the authors' effort on addressing my comments. However, I do not think my concerns/worries are fixed. As the authors could read from my last turn comments, the qualities of the experimental data could not well support their argument. So at that time, I do not recommend it to be published in Nature Communications. Now, even though the authors try to re-analyze their data again for better supporting, but the critical drawback is still there. I would conclude that this is only the preliminary data, not enough on getting the conclusion. Without the solid new data support, I will not recommend it to be published in Nature Communications.

Page 1 of 7
We thank the reviewers for reviewing our manuscript. Below, we respond to the reviewers' comments/questions on the resubmitted manuscript: (Reviewers' comments/questions are in the blue colored italic font and our responses are in the black colored font.) Reviewer #1: The authors have made significant revisions to the manuscript and added new experimental data. Their responses to reviewers' comments are reasonable. Now I recommend its publication in Nature Communications.
We thank the reviewer for reviewing our manuscript and recommending the publication of the revised manuscript in Nature Communications. We are happy to note that the reviewer finds that we have reasonably addressed all the reviewers' comments.
Reviewer #2: Depending on the actual energy spectrum, some gaps are larger than others and experimentally observed more clearly (e.g., colored lines in Fig. R1b). On the other hand, the QHSs observed in TBG are correlated Chern insulator gaps, which occur due to correlation-driven symmetry breaking of four-fold degenerate conduction and valence flat bands (total eight). As depicted in Fig. R1c, conduction and valence Chern bands have Chern numbers of magnitude 1 but with opposite signs and thus explain negative for < 0 and positive for > 0 (Fig. R1d). Figs. R1b and R1d contrast the two physics.   R1b also explains how the dispersions of the Hofstadter gaps in the fan diagram are bound by the rational value of / 0 . This is because at magnetic flux commensurate with the superlattice periodicity, the effective magnetic field felt by electrons is zero resulting in conductivity peaks at / 0 = 1/2, 1/3, 1/4, . .., known as Brown-Zak oscillations. The observation of Brown-Zak oscillations in our data not only corroborates the Hofstadter physics but also explains why some Hofstadter gaps being smaller in size are observed over a narrower range of magnetic fields. To further address the reviewer's comment about the fitting and its uncertainty, we show the analysis of extracting ( , ) of Hofstadter gaps in Fig. R2 in detail. In Fig. R2b, we plot multiple line slices of vs. at different values of and identify the location of minima corresponding to the gaps. We then plot the ( , ) points of the minima in Fig. R2c and fit them with straight lines. As shown in Fig. R2c, the extracted C values indicate low uncertainty in C (for example, the extracted value of the (1, −2) state pointed out by the reviewer comes out to be (0.99, −2.00)). We further note that even for = −2 Chern gaps our fitting is robust across a magnetic field range of 3 T. For references, we include some data in Fig. R3 from other papers on TDBG (mentioned by the reviewer) and TBG, which also report Chern gaps over similar ranges of magnetic field (for example see the (3,1) state in Fig. R3c).

Lastly, based on what is known on the electrostatics of TDBG (DOI:10.1126/science.abc3534) the range of on-site layer energies used in the simulations appears high compared to the electric field used experimentally.
Reply: We appreciate the reviewer's comment. First of all, we would like to emphasize that the values of different hopping parameters used in the band structure calculation of TDBG are not fixed yet and vary across the literature (we use the values from Koshino et al. Phys. Rev. B 99, 235406 (2019)). Therefore, while the theoretical calculation in TDBG helps to understand the underlying physics, an exact quantitative match between experiment and theory is not expected. In our work, the central result that the Hofstadter gaps change their Chern numbers with the electric field is valid for any value of the electric field. When we use slightly higher values of interlayer potential in our calculation, it turns out that even the sequence of the changes in the calculated Chern number matches well with the experiment.
We further add, a relation between the on-site layer potential and the electric displacement field can be written as = / 0 = / 0 . (Eq. R2) Here, is the electron charge, is the distance between graphene layers, and is the dielectric constant. By putting = 0.33 nm and = 4, a typical value of 82.5 meV/(V/nm) is obtained for the factor = / . The mentioned reference also states a similar value of the factor ≈ 59 meV/(V/nm). The conversion, however, depends on the detail variation of which changes with the electric field. For example, the mentioned reference itself shows how the dielectric function can change its value by one order of magnitude as the interlayer potential is increased from 0 to 40 meV (Supplementary Figure S20 in DOI:10.1126/science.abc3534). Additionally, the extent of the screening effect also varies with the twist angle. The variation of screening with the electric field and twist angle could further explain high onsite layer potentials corresponding to the experimental electric field.

Some of the concluding statements are not entirely consistent with the findings. For examples the authors state "In the physics of Chern insulator states, as also in the quantum Hall physics due to the formation of Landau levels, gaps with different Chern numbers can be accessed by changing the Fermi energy by varying the charge density.
In contrast, a pure electrical control, such as the perpendicular electric field, to manipulate the Chern states has not been demonstrated yet." The data shows that one have to change both the electric field and the density in order to change the Chern number. In the calculations the Chern gaps evolve with electric field at a fixed energy. In the experimental data at a fixed magnetic field (flux/moire unit cell) QHS with different Chern numbers are observed at different electric field AND different moire Brillouin zone filling (\nu).

Reply:
We thank the reviewer for this comment. The trajectory of a Chern gap in the parameter space of ( , ) is governed by the Streda formula, Thus, when is varied by changing any parameter (in our experiment ), for a fixed value of , the location of the Chern gap on -axis (equivalently, on -axis) also changes. Therefore, though showing the change in by the electric field at fixed energy is straightforward in the calculation, it involves a change in as well in the experiment. This is depicted schematically in Fig. R4a- Overall the findings are interesting and probably correct, although the experimental data is not entirely convincing.

Reply:
We thank the reviewer for finding our result interesting. We also draw the review's attention to the fact that our experimental results are consistent across multiple devices. We hope that our response establishes that the experimental data convincingly support our findings. Reply: We thank the reviewer for the comment. However, it is not clear to us what data quality the reviewer refers to. The electric field tunability of the Chern gap evolution, which is the central result of our work, is self-evident from Fig. 2a in the main manuscriptthe location of Chern gaps (i.e, dips) clearly changes with the electric field. We further extract the Chern numbers by careful analysisthough all the dips might not be visible in a single color-scale plot, we can clearly track the -minima from the line slices and extract the ( , ) values with low uncertainty as explained in Fig. R2. Furthermore, we would like to draw the reviewer's attention that in the last review response, we did not merely re-analyze old data, but provided new data from the measurement of new devices. That the electric field changes Chern numbers of the Hofstadter gaps is observed across multiple devices and also independently verified by our theoretical calculation.

REVIEWER COMMENTS
Reviewer #2 (Remarks to the Author): I appreciate the authors response, although they may have misunderstood the main question related to the quantum Hall states Chern numbers.
Each broken-symmetry correlated insulator (CI) in either twisted bilayer graphene (TBG) or twisted double bilayer graphene (TDBG) should generate a fan of quantum Hall states with both positive and negative Chern numbers, irrespective of the Chern number of the correlated insulator. However, experimentally the Landau fans tend to point away from charge neutrality in both TBG and TDBG, equivalent to Landau fans have predominantly positive (negative) Chern numbers for positive (negative) moiré unit cell filling. In TBG the CIs also have a non-zero Chern number, but in TBDG the CIs at +-2 electrons per moiré unit cell appear to have a zero Chern number (arXiv:2109.08255). So the experimental observation I was referring to, where Landau fan originating from CIs tend to point away from neutrality is not necessarily dependent on whether the CI is topologically non-trivial or not. In TBG these experimental observations have been theoretically explained by an interaction effect that leads to a smaller (larger) effective mass for excitations away (towards) charge neutrality (Phys. Rev. Lett. 127, 266402, 2021). But experimentally Landau fans in both TBG and TDBG appear to have the same trend, pointing away from neutrality.
That said my statement is based on prior experimental observations, but in principle a quantum Hall state with positive Chern number at negative moiré unit cell filling as reported in the manuscript is not precluded, and I trust the authors have exercised due diligence in data analysis.

Reviewer #3 (Remarks to the Author):
In the first turn, I think I have listed all the details in my review comments. My most concerns are the experimental data cannot support the authors' main claims. In the reply, I did not see any experimental new data added in to improve the paper/data quality. Actually, when I go over another referee's comments on this work, same issue listed there. In the current status, the experimental data is quite preliminary, I am not convinced by their data. Once again, I do not think it is worth to be considered for publication in Nature Communications unless enough and solid experimental data are provided.

Reviewer #4 (Remarks to the Author):
The manuscript by Pratap Chandra Adak et al. reports the study of Chern insulating states in twisted double-bilayer graphene. The authors focus on the Chern insulating states that emerge at high magnetic fields in Hofstadter bands. They use the measurements of the longitudinal resistance as a function of the band filling, displacement field, and magnetic field to make statements regarding these Chern states. While I think some of the data in the paper might be interesting, I can not recommend the paper for publication in Nature Communication for the reasons stated below.
1) The authors emphasize in their abstract that "However, the topological properties of these moiré bands such as Chern numbers are little explored experimentally." I could not agree with this: investigating the topology properties and measuring Chern numbers of the flat moiré bands has been one of the major directions and has already resulted in many exciting discoveries. For example, the discoveries of zero-field Chern insulators (QAHE) in the twisted bilayer and ABC trilayer aligned to hBN, twisted monolayer-bilayer graphene, and recently in twisted TMDs. The Hofstadter Chern insulating states in twisted bilayer were also studied in detail by multiple groups (Ref. 13 and many others).
2) The authors also put an emphasis on the ability to electrically tune Chern states. For example, in line 23: "In contrast, a pure electrical control, such as the perpendicular electric field, to open up a Chern insulating state from a bulk gapless state has not been demonstrated yet." This statement is incorrect. For example, D-field tunable crossings between Landau levels in twisted bilayer graphene were observed ten years ago (PRL 108, 076601 (2012), Sanchez-Yamagishi et al.) Moreover, electric-field-tunable Chern insulators were observed even at zero field, for example, in ABC trilayer on hBN (Chen et al. Nature 579 (2020)) and in twisted monolayer-bilayer graphene (Polshyn et al. Nature 588 (2020)). In the case of ABC on hBN, it was shown that the electrical field alone could be used to switch between C=2 and C=0 states. The authors did not even cite this most relevant paper.
To conclude, I don't see very significant progress or novelty in the results that the author present in their manuscript. The electrical switching of Chern states has already been demonstrated in multiple other systems, and there it happens in more interesting contexts.
3)Finally, I also somewhat agree with the other reviewer regarding the quality of the data. For example, in Fig.2a, half of the marked Chern insulating states are blurred to the extent that they are merging together. Furthermore, the authors say that they could not observe quantization of the Hall conductance for the Chern states explaining it by the disorder. At this point, providing both longitudinal and transverse resistance became the standard in the field. Other groups routinely observe quantized or approximately-quantized Hall resistance for Chern insulating states in both high and zero magnetic fields.
To sum it up, I think that the paper lacks sufficient novelty or a major advance. It also makes several questionable statements. Hence, I cannot recommend the manuscript for publication in Nature Communication. comments/questions: (Reviewers' comments/questions are in blue colored font and our responses are in black colored font.) Reviewer #2 (Remarks to the Author): I appreciate the authors response, although they may have misunderstood the main question related to the quantum Hall states Chern numbers.
Each broken-symmetry correlated insulator (CI) in either twisted bilayer graphene (TBG) or twisted double bilayer graphene (TDBG) should generate a fan of quantum Hall states with both positive and negative Chern numbers, irrespective of the Chern number of the correlated insulator. However, experimentally the Landau fans tend to point away from charge neutrality in both TBG and TDBG, equivalent to Landau fans have predominantly positive (negative) Chern numbers for positive (negative) moiré unit cell filling. In TBG the CIs also have a non-zero Chern number, but in TBDG the CIs at +-2 electrons per moiré unit cell appear to have a zero Chern number (arXiv:2109.08255). So the experimental observation I was referring to, where Landau fan originating from CIs tend to point away from neutrality is not necessarily dependent on whether the CI is topologically non-trivial or not. In TBG these experimental observations have been theoretically explained by an interaction effect that leads to a smaller (larger) effective mass for excitations away (towards) charge neutrality (Phys. Rev. Lett. 127, 266402, 2021). But experimentally Landau fans in both TBG and TDBG appear to have the same trend, pointing away from neutrality.
That said my statement is based on prior experimental observations, but in principle a quantum Hall state with positive Chern number at negative moiré unit cell filling as reported in the manuscript is not precluded, and I trust the authors have exercised due diligence in data analysis.

Reply:
We thank the reviewer for the careful review and for nicely summarizing the trend of Landau levels/Chern insulators in TBG and TDBG. Indeed, unlike usual Landau levels/Chern insulators, the Chern states in the Hofstadter regime evolve in both directions -toward or away from the charge neutrality; we see this in our experiment. This is primarily because energy levels with both positive and negative Chern numbers are intertwined in the fractal of Hofstadter spectra. We see this in our theoretical calculations as well. We also appreciate the reviewer for noting our careful data analysis. Complete understanding of Hofstadter physics in flat bands is ongoing work; we humbly believe that our observations in TDBG will initiate further exploration of electric-field tuning Chern numbers using Hofstadter physics.
In the first turn, I think I have listed all the details in my review comments. My most concerns are the experimental data cannot support the authors' main claims. In the reply, I did not see any experimental new data added in to improve the paper/data quality. Actually, when I go over another referee's comments on this work, same issue listed there. In the current status, the experimental data is quite preliminary, I am not convinced by their data. Once again, I do not think it is worth to be considered for publication in Nature Communications unless enough and solid experimental data are provided.

Reply:
We appreciate the reviewer for his/her valuable time reviewing our manuscript. We would like to point out that not only our experimental results are consistent across multiple devices with twist angles 1.09°, 1.10°, 1.42°, and 1.46° (e.g., see Supplementary Figures 7-9), our independent theoretical calculations firmly support our observations. We have added a new figure in the revised Supplementary Information that shows values mostly consistent with the corresponding Chern numbers. Hence, our observation of minima along with the data provides strong evidence of the ( , ) Chern states. Though the quantization in Hall conductance is somewhat weak, we humbly note that the earlier studies of TDBG (e.g., refs 28-32) have not shown a clear quantization in their magneto-transport data, despite the same groups exploring well-quantized correlated Chern insulator states in TBG. On page 4, while addressing the comments of Reviewer 4, we have further discussed the possible reasons for weak quantization.
Action: In the revised Supplementary Information, we have now included the details of the Hall conductance data ( Supplementary Fig. 5).
Reviewer #4 (Remarks to the Author): The manuscript by Pratap Chandra Adak et al. reports the study of Chern insulating states in twisted double-bilayer graphene. The authors focus on the Chern insulating states that emerge at high magnetic fields in Hofstadter bands. They use the measurements of the longitudinal resistance as a function of the band filling, displacement field, and magnetic field to make statements regarding these Chern states. While I think some of the data in the paper might be interesting, I can not recommend the paper for publication in Nature Communication for the reasons stated below.

Reply:
We thank the reviewer for summarizing our manuscript and appreciating some of the data. Below we address his/her comments.
1) The authors emphasize in their abstract that "However, the topological properties of these moiré bands such as Chern numbers are little explored experimentally." I could not agree with this: investigating the topology properties and measuring Chern numbers of the flat moiré bands has been one of the major directions and has already resulted in many exciting discoveries. For example, the discoveries of zero-field Chern insulators (QAHE) in the twisted bilayer and ABC trilayer aligned to hBN, twisted monolayer-bilayer graphene, and recently in twisted TMDs. The Hofstadter Chern insulating states in twisted bilayer were also studied in detail by multiple groups (Ref. 13 and many others).

Reply:
We thank the reviewer for the comment. We agree with the reviewer that many exciting discoveries including (correlated) Chern Insulator states have resulted from studying different twisted systems (e.g., refs 13-18 in our manuscript). However, we would like to point out that all these works report the observations of Chern insulator states as a consequence of electron correlation. In those studies, the correlation effect facilitates symmetry breaking to lift the degeneracy of four-fold degenerate flat bands. Indeed, without the correlation effect, 'zero-field' Chern insulators cannot arise since Chern insulator states require the breaking of time-reversal symmetry. Otherwise, opposite Chern numbers from two opposite valleys cancel each other.
While observations of correlated Chern insulators are novel due to the possibility of zero-field Chern insulators, an important question remainshow the underlying topology (as indicated by nonzero valley Chern number) of the flat bands manifests without any role of correlations. However, implications of the flat bands' topology in a magnetic field within the framework of singleparticle physics have not received much attention. This is what we meant in the two sentences: "To date, the major focus has been to gain new insights into correlated-electron physics hosted by the flat bands. However, the topological properties of these moiré bands such as Chern numbers are little explored experimentally." Our work explores that direction as we study the implication of underlying band topology on its Hofstadter physics, which does not exclusively require correlations. For example, we demonstrate the perpendicular electric field tunable Chern gaps in devices without (Fig. 2a of the main manuscript, and Supplementary Fig. 7b) and with (Supplementary Fig. 8d and 9b) a correlated insulator feature at v=2. The prospect of interplay with correlations enriches physics further.
Action: In view of the reviewer's comment, to deliver the intended message more appropriately we modified the sentence in the abstract of the revised manuscript: "However, the implications of the topology of these moiré bands within the framework of single-particle physics are little explored experimentally." 2) The authors also put an emphasis on the ability to electrically tune Chern states. For example, in line 23: "In contrast, a pure electrical control, such as the perpendicular electric field, to open up a Chern insulating state from a bulk gapless state has not been demonstrated yet." This statement is incorrect. For example, D-field tunable crossings between Landau levels in twisted bilayer graphene were observed ten years ago (PRL 108, 076601 (2012), Sanchez-Yamagishi et al.) Moreover, electric-field-tunable Chern insulators were observed even at zero field, for example, in ABC trilayer on hBN (Chen et al. Nature 579 (2020)) and in twisted monolayer-bilayer graphene (Polshyn et al. Nature 588 (2020)). In the case of ABC on hBN, it was shown that the electrical field alone could be used to switch between C=2 and C=0 states. The authors did not even cite this most relevant paper.
To conclude, I don't see very significant progress or novelty in the results that the author present in their manuscript. The electrical switching of Chern states has already been demonstrated in multiple other systems, and there it happens in more interesting contexts.

Reply:
We thank the reviewer for the comment. We are aware of the study of D-field tunable Landau level crossings in twisted bilayer graphene by Sanchez-Yamagishi et al. PRL 108, 076601 (2012), which we had cited in the Supplementary Information. Previous works from our group also explored the physics of Landau level crossing in ABA-trilayer graphene in detail (Nat. Comm. 8, 14518 (2017), PRL 121, 056801 (2018)). However, Landau level crossings do not necessarily change the Chern number (these crossings occur between Landau levels with the same Chern number). For example, Sanchez-Yamagishi et al. studied the change in layer polarization to spinpolarization state by Landau level crossing due to D-field; there was no change in Chern number as a function of D-field. Similar work was done in bilayer graphene (BLG) as well (e.g., Science 330, 812 (2012)).
While the work by Polshyn et al. Nature 588, 66 (2020) explored the electrical control of Chern states, the control was achieved by changing the charge density, not the pure electric field. We appreciate the reviewer for highlighting the relevance of the important work by Chen et al. Nature 579, 56 (2020). While the electric field tunability of correlated Chern insulator states studied by Chen et al. is novel, we believe that our demonstration is also significant for not requiring correlations as a prerequisite. Furthermore, the electric field tunability of Chern numbers is abundant in our system since Hofstadter spectra naturally possess plenty of subbands with many different Chern numbers.

Action:
In the revised manuscript, we have included the mentioned references in appropriate locations. To further clarify the novel findings in our work, we have added the following sentences in the discussion of the revised manuscript: "Here we note that the control over Chern states has been recently demonstrated in twisted monolayer-bilayer graphene by tuning the charge density and in hBN-aligned ABC trilayer graphene using the electric field. Our work demonstrates a novel pathway to control Chern states using the electric field without requiring electron correlation. Furthermore, the Hofstadter platform of TDBG offers a plethora of Chern transitions over a broad region of electric field." 3) Finally, I also somewhat agree with the other reviewer regarding the quality of the data. For example, in Fig.2a, half of the marked Chern insulating states are blurred to the extent that they are merging together. Furthermore, the authors say that they could not observe quantization of the Hall conductance for the Chern states explaining it by the disorder. At this point, providing both longitudinal and transverse resistance became the standard in the field. Other groups routinely observe quantized or approximately-quantized Hall resistance for Chern insulating states in both high and zero magnetic fields.

Reply:
We thank the reviewer for the comment. In Fig. 2a, we plot the longitudinal conductance in a color-scale plot to show the plethora of tunable Chern states. Though it helps to visualize multiple Chern states all at once, we agree that some individual states look blurred. However, as we showed in Supplementary Fig. 6, the minima corresponding to Chern gaps are prominent in the line slices and we can trace them over a large range of magnetic field to extract the corresponding Chern numbers unambiguously.
In Fig. R1, we plot the Hall conductance as a function of the filling across Chern gaps. We do observe approximately-quantized Hall conductance. For example, the measured values of at the location of minima corresponding to different Chern gaps (shaded with blue-colored backgrounds) are very close to the quantized value of 2 /ℎ (indicated by dashed lines).  The absence of a clear quantization in the Hall conductance from the Hofstadter butterfly can be understood by the low magnitude of the Hofstadter gaps. As revealed by our temperaturedependent transport, even the most prominent Hofstadter gaps have magnitudes of 0.1 to 0.4 meV (see Supplementary Fig. 2). Such low values of the gap can be due to the following reasons: 6 • Hofstadter physics generically involves gaps with low magnitude, because each Landau level in Hofstadter spectra actually splits into numerous subbands (fractal). For example, quantized QHSs were observed in graphene even at room temperature (Novoselov et. al., Science 335, 1379, 2007. In contrast, the observation of Hofstadter butterflies in graphene/hBN required a low temperature and high magnetic field (Hunt et al., Science 340, 6139, 2013). Indeed, unlike Hofstadter physics, the physics of correlated Chern insulator states involve only a few bands: correlated Chern insulator states occur when the correlation effect separates otherwise degenerate four flat Chern bands. • The low bandwidth of the underlying flat bands in twisted double bilayer graphene further minimizes the energy scale. We discussed this aspect in Supplementary Note 6 and Supplementary Fig. 11. • The opposite values of Chern numbers from two valleys dictate opposite evolutions of the corresponding Hofstadter spectra. As discussed in Fig. 3 of the main manuscript, this can result in energy gaps in the K-valley spectrum filled by the energy levels of the K'-valley spectrum (and vice versa). Thus, the nontrivial topology of the zero-magnetic field band structure further obscures the Hofstadter gaps.
On top of the low energy gaps, the twist-angle inhomogeneity can further impede the observation of quantization. While angle disorder is a common problem in the twistronics field, we would like to note that our devices are of state-of-the-art quality as reflected by the observation of correlated insulator gaps at zero magnetic field (see Supplementary Fig. 8a and 8b).
Finally, we note some interesting facts from the literature which somewhat contradict the reviewer's assessment about routine observation of quantization by other groups, especially in TDBG.
• The earlier experimental reports on twisted double bilayer graphene do not show a clear quantization in their magneto-transport data though they routinely assign Chern numbers solely by locating the minima in longitudinal resistance/conductance. For example, in Fig.  R2, we put magneto-transport data from some novel reports that explore correlation physics in twisted double bilayer graphene. They do not show corresponding Hall conductance with clear quantization. Interestingly, often the same groups report quantization in the observation of correlated Chern insulator states in TBG. This corroborates that the energy scales play a greater role than the device quality in weakening the quantization. • The first report on observing the giant anomalous Hall (AH) effect in TBG by Sharpe et al.
Science 365, 605 (2019) notes in their abstract, "the AH resistance is not quantized, and dissipation is present…." However, Serlin et al. Science 367, 900 (2020) observed an "intrinsic quantized anomalous Hall effect' in the same system later. • Despite the observation of quantized AH states in TBG, as noted in the last point, none of the two reports on AH states in TDBG shows quantization (He et al. arXiv:2109.08255, Kuiri et al. arXiv:2204). In fact, arXiv:2109.08255, to which Reviewer 2 also referred, states: "The amplitude of the AHE is small in device O1, much less than the quantized value of ℎ/ 2 ." While a very recent work by Liu et al. Nat. Comm. 13, 3292 (2022) reports quantization, the quantization in TDBG devices seems elusive in general. In essence, our observation of minima and approximate quantization in provide evidence of the ( , ) Chern states.

Action:
In the revised Supplementary Information, we have now included the details of the Hall conductance data (Supplementary Fig. 5).
To sum it up, I think that the paper lacks sufficient novelty or a major advance. It also makes several questionable statements. Hence, I cannot recommend the manuscript for publication in Nature Communication.

Reply:
We thank the reviewer for assessing our manuscript. However, we believe that our detailed response carefully addresses the possible issues. In summary, while small energy scales together with twist-angle inhomogeneity can hinder robust quantization in complex Hofstadter fractals in TDBG, our independent theoretical calculations further validate the experimental observation. In literature, the initial reports observing novel quantum states often lack robust quantization, however, they open up new directions. Similarly, we believe that our work will initiate further exploration of electric-field tuning of Chern numbers using Hofstadter physics.
I appreciate the effort of the authors to address my comments. However, I am afraid that my concerns remain the same. Most of the features that the authors claim to be novel in the introduction (e.g., opening a gap with an electric field) are not new. The original finding of the paper -the observed evolution of the Chern states with the perpendicular electric field and carrier density is curious but is not of high significance to the field.
1. Regarding the author's response to my comment #2: even after modification, I am afraid I still have to disagree with their statement "Recently a pure electrical control, such as the perpendicular electric field, to open up a Chern insulating state from a bulk gapless state has been demonstrated using a correlated system. Similar electrical control over Chern states without requiring electron correlation will be novel." Opening up a Chern insulating state from a gapless state using an electric field is exactly what was demonstrated in the ten-year-old paper that I mentioned in my comment (PRL 108, 076601 (2012), Sanchez-Yamagishi et al.) and hence could hardly be considered novel. The displacement field allows one to tune between compressible states (when Landau levels cross) and incompressible states with finite Chern numbers, as is evident from Fig 2f in that paper. Moreover, in that case, the electronic correlations were indeed negligible in contrast to TDBG.
2. I want to discuss this last point further. Despite the authors repeatedly emphasize that their observations are unique because, as they claim, they do not require strong electronic correlations. The problem with this is that the systems that they study are still inherently strongly-correlated. For example, Cao et al. Nature 2020, observed correlated states in devices with nearly the same twist angle 1.1 deg. Moreover, the absence of a clear insulting state at half-filings is not evidence of the absence of correlations. It is common to observe interaction-driven metallic states with reduced spin and valley degeneracy that show up as a "halo" in Rxx even without robust insulating states. For example, Shen et al. Nat Phys. 2020, observed this in 1.06deg TDBG devices. Authors themselves commented that they see such halo regions. Hence, it is very likely that the correlations in their devices are, in fact, strong and they will considerably affect the band structure and phenomenology of the system. I also want to note that the authors themselves invoke strong correlations in flat bands to explain strong spin splitting in their model (line 133). In such case, speculations that the observed physics doesn't require correlations seems to me unnecessary and even misleading.